The back-reaction of baryons on the dark matter halo density profile is of great interest, not least because it is an important systematic uncertainty when attempting to detect the dark matter. Here, we draw on a large suite of high-resolution cosmological hydrodynamical simulations to systematically investigate this process and its dependence on the baryonic physics associated with galaxy formation. The effects of baryons on the dark matter distribution are typically not well described by adiabatic contraction models. In the inner 10per cent of the virial radius the models are only successful if we allow their parameters to vary with baryonic physics, halo mass and redshift, thereby removing all predictive power. On larger scales the profiles from dark matter only simulations consistently provide better fits than adiabatic contraction models, even when we allow the parameters of the latter models to vary. The inclusion of baryons results in significantly more concentrated density profiles if radiative cooling is efficient and feedback is weak. The dark matter halo concentration can in that case increase by as much as 30 (10) per cent on galaxy (cluster) scales. The most significant effects occur in galaxies at high redshift, where there is a strong anticorrelation between the baryon fraction in the halo centre and the inner slope of both the total and the dark matter density profiles. If feedback is weak, isothermal inner profiles form, in agreement with observations of massive, early-type galaxies. However, we find that active galactic nuclei (AGN) feedback, or extremely efficient feedback from massive stars, is necessary to match observed stellar fractions in groups and clusters, as well as to keep the maximum circular velocity similar to the virial velocity as observed for disc galaxies. These strong feedback models reduce the baryon fraction in galaxies by a factor of 3 relative to the case with no feedback. The AGN is even capable of reducing the baryon fraction by a factor of 2 in the inner region of group and cluster haloes. This in turn results in inner density profiles which are typically shallower than isothermal and the halo concentrations tend to be lower than in the absence of baryons. We therefore conclude that the disagreement between the concentrations inferred from observations of groups of galaxies and predictions from simulations that was identified by Duffy et al. is not alleviated by the inclusion of baryons.